Optimal. Leaf size=95 \[ \frac{b x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a (b c-a d)}-\frac{d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)} \]
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Rubi [A] time = 0.105637, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{b x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a (b c-a d)}-\frac{d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^n)^(-1 - n^(-1))/(a + b*x^n),x]
[Out]
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Rubi in Sympy [A] time = 13.6199, size = 70, normalized size = 0.74 \[ \frac{d x \left (c + d x^{n}\right )^{- \frac{1}{n}}}{c \left (a d - b c\right )} - \frac{b x \left (c + d x^{n}\right )^{- \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, 1 \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (- a d + b c\right )}{a \left (c + d x^{n}\right )}} \right )}}{a \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*x**n)**(-1-1/n)/(a+b*x**n),x)
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Mathematica [C] time = 2.23558, size = 153, normalized size = 1.61 \[ \frac{x \left (c+d x^n\right )^{-\frac{n+1}{n}} \left (\frac{b n x^{2 n} (a d-b c) \, _2F_1\left (2,2+\frac{1}{n};3+\frac{1}{n};\frac{(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a^2 (2 n+1) \left (c+d x^n\right )}+\frac{b x^n \Phi \left (\frac{(a d-b c) x^n}{a \left (d x^n+c\right )},1,1+\frac{1}{n}\right )}{a}+\frac{a \left (c+d x^n\right )}{c \left (a+b x^n\right )}\right )}{a} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^n)^(-1 - n^(-1))/(a + b*x^n),x]
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Maple [F] time = 0.126, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( c+d{x}^{n} \right ) ^{-1-{n}^{-1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*x^n)^(-1-1/n)/(a+b*x^n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n - 1)/(b*x^n + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}^{\frac{n + 1}{n}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n - 1)/(b*x^n + a),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d*x**n)**(-1-1/n)/(a+b*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}^{-\frac{1}{n} - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n - 1)/(b*x^n + a),x, algorithm="giac")
[Out]